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The Laplace Equa­tion is a homo­gen­eous dif­fer­en­tial equa­tion which is seen all over phys­ics. For this reas­on, its solu­tions are of great importance.

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Video # Video Tutori­al Title Remarks
1 Intro­duc­tion to Laplace’s Equation V” = 0
2 Earnshaw’s The­or­em No loc­al max­ima or min­ima are permitted
3 First Unique­ness Theorem Why the function’s value at the bound­ary is important
4 Second Unique­ness Theorem  
5 Intro­duc­tion to Sep­ar­a­tion of Variables The fun­da­ment­als of using this very power­ful tech­nique to solve dif­fer­en­tial equations
6 Char­ac­ter­ist­ic Equa­tion 2/2 Deriv­a­tion of the Char­ac­ter­ist­ic Equation
7 Sep­ar­a­tion of Vari­ables Example 1  V” = 0 to include the use of Four­i­er Series
8 Sep­ar­a­tion of Vari­ables Example 2  V” = 0 (new bound­ary con­di­tions) to include the use of Four­i­er Series
9 Sep­ar­a­tion of Vari­ables Example 3  V” = 0 (new bound­ary con­di­tions) to include the use of Four­i­er Series
10 Prop­er­ties of Sep­ar­able Solutions  Why sta­tion­ary state solu­tions are the fun­da­ment­als of Quantum Mechanics
11 Spher­ic­al Polar Co-ordinates  Rect­an­gu­lar to spher­ic­al polar co-ordin­ates is required for the wave­func­tions of Hydrogen
12 Laplace Equa­tion in Spher­ic­al Co-ords  Con­vert­ing the D.E., to spher­ic­al polar coordinates
13 The Radi­al Equation The first of the three ordin­ary equa­tions res­ult­ing from the meth­od of sep­ar­a­tion of variables
14 Lapla­cian in Spher­ic­al Coordinates A very tedi­ous deriv­a­tion of a very import­ant operator
15 The Azi­muth­al Equation The second of the three ordin­ary equa­tions res­ult­ing from the meth­od of sep­ar­a­tion of variables
16 The Polar Angle Equation The third of the three ordin­ary equa­tions res­ult­ing from the meth­od of sep­ar­a­tion of variables

 

 

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